Elect. Comm. in Probab. 12 (2007), 259–266
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
A NOTE ON ERGODIC TRANSFORMATIONS OF
SELF-SIMILAR VOLTERRA GAUSSIAN PROCESSES
CÉLINE JOST
Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hällströmin katu 2b), 00014
University of Helsinki, Finland
email: celine.jost@iki.fi
Submitted February 14, 2007, accepted in final form July 30, 2007
AMS 2000 Subject classification: 60G15; 60G18; 37A25
Keywords: Volterra Gaussian process; Self-similar process; Ergodic transformation; Fractional
Brownian motion
Abstract
We derive a class ofRergodic transformations of self-similar Gaussian processes that are Volterra,
t
i.e. of type Xt = 0 zX (t, s)dWs , t ∈ [0, ∞), where zX is a deterministic kernel and W is a
standard Brownian motion.
1
Introduction
Let (Xt )t∈[0,∞) be a continuous Volterra Gaussian process on a complete probability space
(Ω, F, P). This means that
Xt =
Z
t
0
zX (t, s)dWs , a.s., t ∈ [0, ∞),
(1.1)
where the kernel zX ∈ L2loc [0, ∞)2 is Volterra, i.e. zX (t, s) = 0, s ≥ t, and (Wt )t∈[0,∞) is a
standard Brownian motion. Clearly, X is centered and
Z s
X
R (s, t) := CovP (Xs , Xt ) =
zX (s, u)zX (t, u)du, 0 ≤ s ≤ t < ∞.
0
We assume that X is β-self-similar for some β > 0, i.e.
d
(Xat )t∈[0,∞) =
d
aβ X t
t∈[0,∞)
, a > 0,
(1.2)
where = denotes equality of finite-dimensional distributions. Furthermore, we assume that zX
is non-degenerate in the sense that the family {zX (t, ·) | t ∈ (0, ∞)} is linearly independent
and generates a dense subspace of L2 ([0, ∞)). Then
Γt (X) := span{Xs | s ∈ [0, t]} = Γt (W ), t ∈ (0, ∞),
259
(1.3)
260
Electronic Communications in Probability
where the closure is in L2 (P), or equivalently,
FX = FW ,
where FX := FtX
t∈[0,∞)
denotes the completed natural filtration of X.
We assume implicitly that (Ω, F, P) is the coordinate space of X, which means that Ω =
X
{ω : [0, ∞) → R | ω is continuous}, F = F∞
:= σ(Xt | t ∈ [0, ∞)) and P is the probability
measure with respect to which the coordinate process Xt (ω) = ω(t), ω ∈ Ω, t ∈ [0, ∞), is a
centered Gaussian process with covariance function RX . Recall that a measurable map
Z : (Ω, F, P) → (Ω, F, P)
X(ω)
7→ Z(X(ω))
is a measure-preserving transformation, or endomorphism, on (Ω, F, P) if PZ = P, or equid
valently, if Z(X) = X. If Z is also bijective and Z −1 is measurable, then it is an automorphism.
Processes of the above type are a natural generalization of the nowadays in connection with
finance and telecommunications extensively studied fractional
Brownian motion with Hurst
index H ∈ (0, 1), or H-fBm. The H-fBm, denoted by BtH t∈[0,∞) , is the continuous, centered
Gaussian process with covariance function
H
RB (s, t) =
1 2H
s
+ t2H − |s − t|2H , s, t ∈ [0, ∞).
2
For H = 12 , fBm is standard Brownian motion. H-fBm is H-self-similar and has stationary
increments. The non-degenerate Volterra kernel is given by
1
1
1
1
t
, 0 < s < t < ∞,
zB H (t, s) = c(H)(t − s)H− 2 · 2 F1
− H, H − , H + , 1 −
2
2
2
s
where c(H) :=
2HΓ( 23 −H )
Γ(H+ 12 )Γ(2−2H)
21
with Γ denoting the Gamma function, and 2 F1 is the
Gauss hypergeometric function. In 2003, Molchan (see [8]) showed that the transformation
Z t H
Bs
ds, t ∈ [0, ∞),
(1.4)
Zt B H := BtH − 2H
s
0
is measure-preserving and satisfies
ΓT Z B H
where
= ΓT Y H , T > 0,
YtH := MtH −
Here, MtH :=
RT
ξTH := 2H 0
t H
ξ , t ∈ [0, T ].
T T
(1.5)
(1.6)
√
Rt 1
2 − 2H 0 s 2 −H dWs , t ∈ [0, ∞), is the fundamental martingale of B H and
s 2H−1
dMsH .
T
In this work, we present a class of measure-preserving transformations (on the coordinate
space) of X, which generalizes this result. Moreover, we show that these measure-preserving
transformations are ergodic.
A note on ergodic transformations of
self-similar Volterra Gaussian processes
2
261
Ergodic transformations
First, we introduce the class of measure-preserving transformations:
Theorem 2.1. Let α >
−1
2 .
Then the transformation
1
Ztα (X) := Xt − (2α + 1)tβ−α− 2
Z
0
t
1
sα−β− 2 Xs ds, t ∈ [0, ∞),
is an automorphism on the coordinate space of X. The inverse is given by
Z ∞
1
3
Ztα,−1 (X) = Xt − (2α + 1)tα+β+ 2
Xs s−β−α− 2 ds, a.s., t ∈ [0, ∞).
(2.1)
(2.2)
t
The integrals on the right-hand sides are L2 (P)-limits of Riemann sums.
Proof. First, note that RX is continuous. Furthermore, by combining Hölder’s inequality and
(1.2), we have that
RX (s, t) ≤ EP (X1 )2 sβ tβ , s, t ∈ (0, ∞).
Hence, the double Riemann integrals
Z tZ t
0
and
Z
t
∞
1
(us)α−β− 2 RX (u, s)duds
0
Z
∞
3
(us)−β−α− 2 RX (u, s)duds
t
are finite. Thus, the integrals in (2.1) and (2.2) are well-defined (see [5], section 1).
Second, we show that Z α is a measure-preserving transformation. Let Yt := exp(−βt)Xexp(t)
α
and Ytα := exp(−βt)Zexp(t)
(X), t ∈ R, denote the Lamperti transforms of X and Z α (X),
respectively. Hence, the process (Yt )t∈R is stationary. By substituting v := ln(s), we obtain
that
!
Z exp(t)
1
1
Ytα = exp(−βt) Xexp(t) − (2α + 1) exp
β−α−
t
sα−β− 2 Xs ds
2
0
Z t
1
1
= Yt − (2α + 1) exp
−α −
t
Xexp(v) dv
exp v α − β +
2
2
−∞
Z t
1
1
t
Yv dv
exp v α +
= Yt − (2α + 1) exp
−α −
2
2
−∞
Z ∞
=
hα (t − v)Yv dv, a.s., t ∈ R,
−∞
where
1
x , x ∈ R.
hα (x) := δ0 (x) − (2α + 1)1(0,∞) (x) exp − α +
2
Thus, Y α is a linear, non-anticipative, time-invariant transformation of Y . The spectral distribution function of Y α is given by (see [13], p. 151)
2
dF α (λ) = H α (λ) dF (λ), λ ∈ R,
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Electronic Communications in Probability
R
where H α (λ) := R exp(−iλx)hα (x)dx denotes the Fourier transform of hα and F is the
spectral distribution function of Y . We have that (see [3], p. 14 and p. 72)
!
1
− 21 − α + iλ
−
iλ
α
+
2
=
= 1, λ ∈ R,
H α (λ) = 1 − (2α + 1)
2
1
1
+ α + λ2
2 + α + iλ
2
d
d
i.e. F α ≡ F . It follows from this that (Ytα )t∈R = (Yt )t∈R , or equivalently, (Ztα (X))t∈[0,∞) =
(Xt )t∈[0,∞) .
Third, by splitting integrals and using Fubini’s theorem, we obtain that
Ztα,−1 (Z α (X)) = Xt = Ztα Z α,−1 (X) , a.s., t ∈ [0, ∞).
1
Remark 2.2. Theorem 2.1 generalizes (1.4). In fact, Z H− 2 B H = Z B H , H ∈ (0, 1).
Remark 2.3. Theorem 2.1 holds true for general continuous centered β-self-similar Gaussian
processes.
Next, we present two auxiliary lemmas concerning the structure of zX :
Lemma 2.4. Let (Xt )t∈[0,∞) be a Volterra Gaussian process with a non-degenerate Volterra
kernel zX . Then the following are equivalent:
1. X is β-self-similar, i.e.
Z s
Z s
2β−1
zX (at, au)zX (as, au)du = a
zX (t, u)zX (s, u)du, 0 < s ≤ t < ∞, a > 0.
0
0
2. It holds that
1
3. There exists FX
zX (at, as) = aβ− 2 zX (t, s), 0 < s < t < ∞, a > 0.
∈ L2 (0, 1), (1 − x)2β−1 dx such that
s
1
, 0 < s < t < ∞.
zX (t, s) = (t − s)β− 2 FX
t
1
Proof. 1 ⇒ 2: For a > 0, let zY (a) (t, s) := a 2 −β zX (at, as), 0 < s < t < ∞, and let Yt (a) :=
Rt
0 zY (a) (t, s)dWs , t ∈ [0, ∞). Clearly, zY (a) is non-degenerate. From (1.3), we obtain that
d
Γ (Y (a)) = Γt (W ), t ∈ (0, ∞). From part 1, it follows that X = Y (a). Thus, the process Wt′ :=
R tt ∗
∗
0 zX (t, s)dYs (a), t ∈ [0, ∞), where zX is the reciprocal of zX and the integral is an abstract
Wiener integral, is a standard Brownian motion with Γt (W ′ ) = Γt (Y (a)), t ∈ (0, ∞). Hence,
Rt
Γt (W ) = Γt (W ′ ), i.e. W and W ′ are indistinguishable. Therefore, Yt (a) = 0 zX (t, s)dWs ,
a.s., t ∈ (0, ∞), i.e. Yt (a) = Xt , a.s., t ∈ [0, ∞). In particular, 0 = EP (Yt (a) − Xt )2 =
2
Rt
zY (a) (t, s) − zX (t, s) ds, t ∈ (0, ∞). Thus, zX (t, ·) ≡ zY (a) (t, ·), t ∈ (0, ∞).
0
1
2 ⇒ 3: Let GX (t, s) := (t − s) 2 −β zX (t, s), 0 < s < t < ∞. From part 2, it follows that
GX (at, as) = GX (t, s), 0 < s < t < ∞, a > 0. Hence, for every (t, s), s < t, the function
s
GX is constant on
depends only on the
the line {(at, as) | a ∈ (0, ∞)}, which
slope t . Thus,
s
2
2β−1
dx .
GX (t, s) = FX t , 0 < s < t < ∞, for some FX ∈ L (0, 1), (1 − x)
3 ⇒ 1: This is trivial.
Lemma 2.5. Let α > −1
2 . Then we have that
Z t
Z t
1
1
tβ−α− 2
uα−β− 2 zX (u, s)du = sα
zX (t, u)u−α−1 du, 0 < s < t < ∞.
s
s
A note on ergodic transformations of
self-similar Volterra Gaussian processes
263
Proof. From Lemma 2.4, it follows that the Volterra kernel of X can be written as
s
1
, 0 < s < t < ∞,
zX (t, s) = (t − s)β− 2 FX
t
for some function FX . By substituting first x := us and then v := tx, we obtain that
Z t
Z t
s
1
1
1
1
1
uα−β− 2 zX (u, s)du = tβ−α− 2
uα−β− 2 (u − s)β− 2 FX
du
tβ−α− 2
u
s
s
Z 1
1
1
x−α−1 (1 − x)β− 2 FX (x)dx
= tβ−α− 2 sα
s
t
sα
=
Z
t
s
sα
=
Z
1
(t − v)β− 2 FX
t
v t
v −α−1 dv
zX (t, v)v −α−1 dv.
s
The next lemma is the key result for deriving the ergodicity of the measure-preserving transformations:
Lemma 2.6. Let α >
−1
2 .
Then
Ztα (X)
Z
=
t
0
zX (t, s)dZsα (W ), a.s., t ∈ [0, ∞).
Proof. By combining (1.1) and the stochastic Fubini theorem, using Lemma 2.5, again the
stochastic Fubini theorem, and finally using partial integration, we obtain that
Z t
Z t
β−α− 21
α
α−β− 12
t
Zt (X) = Xt − (2α + 1)
zX (s, u)ds dWu
s
0
u
Z t Z t
α
−α−1
u
zX (t, s)s
ds dWu
= Xt − (2α + 1)
0
u
t
Z s
zX (t, s)s−α−1
uα dWu ds
0
0
Z t
Z s
= Xt − (2α + 1)
zX (t, s) (−α)s−α−1
uα−1 Wu duds + s−1 Ws ds
= Xt − (2α + 1)
Z
0
=
Z
0
t
0
zX (t, s)dZsα (W ), a.s., t ∈ (0, ∞).
n
In the following, let Z α,n := (Z α ) denote the n-th iterate of Z α , n ∈ Z. Also, let
Γ∞ (X) := span{Xt | t ∈ [0, ∞)}.
For α > − 12 , let
Ntα
:=
Z
0
Clearly, N α is an α +
1
2
t
sα dWs , t ∈ [0, ∞).
-self-similar FX -martingale. From Lemma 2.6, it follows that
Z t
α
Zt (X) =
zX (t, s)s−α dZsα (N α ) , a.s., t ∈ (0, ∞).
(2.3)
0
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Electronic Communications in Probability
The next lemma is an auxiliary result, which was obtained in [7], section 3.2 and Theorem
5.2. (The automorphism Z α on the coordinate space of N α here corresponds to the (ergodic)
automorphism T (1) on the coordinate space of the martingale M with M := N α in [7].)
Lemma 2.7. Let α >
1. It holds that
−1
2
and T > 0.
ΓT (Z α (N α )) = ΓT N α,T ,
2α+1 α
NT , t ∈ [0, T ], is a bridge of N α , i.e. a process satisfying
where Ntα,T := Ntα − Tt
LawP N α,T = LawP (N α | NTα = 0), and ΓT N α,T := span Ntα,T | t ∈ [0, T ] .
2. We have that
(2.4)
ΓT (N α ) = ⊥n∈N0 span ZTα,n N α
and
Γ∞ (N α ) = ⊥n∈Z span ZTα,n N α .
(2.5)
Here, ⊥ denotes the orthogonal direct sum.
By combining (2.3) and part 1 of Lemma 2.7, we obtain the following:
Lemma 2.8. Let α >
−1
2
and T > 0. Then
ΓT (Z α (X)) = ΓT N α,T .
Remark 2.9. Lemma 2.8 is a generalization of identity (1.5). Indeed, we have that YtH =
√
Rt
H− 1 ,T
2 − 2H 0 s1−2H dNs 2 , a.s., t ∈ [0, T ], where Y H is the process defined in (1.6). Y H is a
bridge (of some process) if and only if H = 21 .
The following generalizes part 2 of Lemma 2.7:
Lemma 2.10. Let α >
−1
2
and T > 0. Then we have that
ΓT (X) = ⊕n∈N0 span {ZTα,n (X)}
and
Γ∞ (X) = ⊕n∈Z span {ZTα,n (X)}.
Proof. We assume that X 6= N α . By iterating (2.3) and (2.4), we obtain that
n
o
= ⊥i≥n span ZTα,i N α , n ∈ Z.
ZTα,n (X) ∈ ΓT (Z α,n (X)) = ΓT Z α,n N α
Moreover, XT 6⊥ NTα , hence ZTα,n (X) 6⊥ ZTα,n (N α ), n ∈ Z, and therefore,
n
o
ZTα,n (X) 6∈ ⊥i≥n+1 span ZTα,i (N α ) , n ∈ Z.
From (2.4) and (2.5), it follows that the systems {ZTα,n (X)}n∈N0 and {ZTα,n (X)}n∈Z are free
and complete in ΓT (X) and Γ∞ (X), respectively.
Remark 2.11. The process X is an FX -Markov process if and only if there exists α > −1
2
and a constant c(X), such that
Z t
1
sα dWs , a.s., t ∈ (0, ∞).
(2.6)
Xt = c(X) · tβ− 2 −α
0
The free complete system
{ZTα,n (X)}n∈Z
is orthogonal if and only if (2.6) is satisfied.
A note on ergodic transformations of
self-similar Volterra Gaussian processes
265
From Lemma 2.10, we obtain the following:
Corollary 2.12. Let α >
−1
2
and T > 0. Then
FTX =
_
n∈N0
σ ZTα,n (X) .
Furthermore,
X
F = F∞
=
_
n∈Z
σ ZTα,n (X) .
Recall that an automorphism Z is a Kolmogorov automorphism, if there exists a σ-algebra
A ⊆ F , such that Z −1 A ⊆ A, ∨m∈Z Z m A = F and ∩m∈N0 Z −m A = {Ω, ∅}. A Kolmogorov
automorphism is strongly mixing and hence ergodic (see [11], Propositions 5.11 and 5.9 on p.
63 and p. 62). The ergodicity of Z α is hence a consequence of the following:
Theorem 2.13. Let α > −1
Z α and Z α,−1 are Kolmogorov
2 and T > 0. The
automorphisms
α,n
X
automorphisms with A = ∨n∈−N σ ZT (X) and A = FT , respectively.
Proof. Z α is a Kolmogorov automorphism with A = ∨n∈−N σ ZTα,n (X) :
First, Z α,−1 A = ∨n∈−N σ ZTα,n−1 (X) ⊆ A.
Second, ∨m∈Z Z α,m A = ∨m∈Z ∨n∈−N σ ZTα,m+n (X) = F.
Third, let {Yn }n∈−N denote the Hilbert basis of ⊕n∈−N span {ZTα,n (X)} which is obtained
from {ZTα,n (X)}n∈−N via Gram-Schmidt orthonormalization. By using Kolmogorov’s zero
α,−m
A = ∩m∈N0 ∨n≤−m−1 σ ZTα,n (X) =
one law (see [12], p. 381),
we obtain that ∩m∈N0 Z
∩m∈N0 ∨n≤−m−1 σ Yn = {Ω, ∅}.
Similarly, one shows that Z α,−1 is a Kolmogorov automorphism with A = FTX .
Acknowledgements. Thanks are due to my supervisor Esko Valkeila and to Ilkka Norros
for helpful comments. I am indebted to the Finnish Graduate School in Stochastics (FGSS)
and the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisäla Foundation
for financial support.
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